Small ball probabilities for the stochastic heat equation with colored noise
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929447262748672 |
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| author | Chen, Jiaming |
| author_facet | Chen, Jiaming |
| contents | We consider the stochastic heat equation on the 1-dimensional torus $\mathbb{T}:=\left[-1,1\right]$ with periodic boundary conditions: $$ \partial_t u(t,x)=\partial^2_x u(t,x)+σ(t,x,u)\dot{F}(t,x),\quad x\in \mathbb{T},t\in\mathbb{R}_+, $$ where $\dot{F}(t,x)$ is a generalized Gaussian noise, which is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_04534 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Small ball probabilities for the stochastic heat equation with colored noise Chen, Jiaming Probability Primary, 60H15, Secondary, 60G60 We consider the stochastic heat equation on the 1-dimensional torus $\mathbb{T}:=\left[-1,1\right]$ with periodic boundary conditions: $$ \partial_t u(t,x)=\partial^2_x u(t,x)+σ(t,x,u)\dot{F}(t,x),\quad x\in \mathbb{T},t\in\mathbb{R}_+, $$ where $\dot{F}(t,x)$ is a generalized Gaussian noise, which is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$. |
| title | Small ball probabilities for the stochastic heat equation with colored noise |
| topic | Probability Primary, 60H15, Secondary, 60G60 |
| url | https://arxiv.org/abs/2202.04534 |